Parametric modelling in motion and design
Введено| English | Значение |
|---|---|
| parametric model/ˌpærəˈmetrɪk ˈmɒdl/ | параметрическая модель |
When does a mathematical path stop describing a real flight?
- A ball’s height and horizontal position depend on the same time. A Cartesian path cannot tell us when it is reached without the time model.
- This lesson studies parametric model 参数模型: A model expressing two quantities using a common variable and stated assumptions.
The method
- Name the parameter, its units and allowed interval.
- Evaluate both model quantities at the same parameter value.
- Eliminate the parameter to relate the quantities, then interpret endpoints and limitations.
- A mathematical curve beyond the allowed interval need not describe the physical situation.
The flight model y=20t-5t² gives negative heights for t>4. What does that mean physically?
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
A model is valid on its stated interval, here 0≤t≤4. Outside it, the mathematics continues but the physics does not - interpreting endpoints is part of the modelling skill.
A worked case
A simplified flight model uses x=6t and y=20t−5t² in metres, with t in seconds and 0≤t≤4. It assumes constant horizontal speed and downward acceleration 10 metres per second squared, ignoring air resistance. The noninitial ground time is t=4, giving x=24 m. Completing the square gives y=20−5(t−2)², so maximum height is 20 m at t=2. Eliminating t gives y=10x/3−5x²/36 for 0≤x≤24. For a rectangle of width t and length t+2 metres, P=4t+4 and A=t(t+2). With 0<t≤5, A=(P²−16)/16 and 4<P≤24.
Parametric modelling in motion and design
Name the parameter, its units and allowed interval
Identify which statements preserve the original parameter, curve segment and contextual assumptions.
Find the horizontal ground distance in the stated flight model.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
y=t(20−5t)=0 gives t=4 after launch; x=6×4=24 m.
Where it goes wrong
- The flight formula giving negative height after t=4 does not describe continued free flight below the ground.
- Do not use t measured in minutes with coefficients defined for seconds.
- In the rectangle model, t=0 is excluded because the width must be positive.
Not true: The Cartesian flight curve is a physically valid model for every real x. The model applies only while the object is in flight, between launch and landing.
Find the maximum height in the same model.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Complete the square: y=20−5(t−2)², with maximum 20 m.
The Cartesian flight curve is a physically valid model for every real x.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
The model applies only while the object is in flight, between launch and landing.
Applying it
- Check a model against observed data and its assumptions.
- Wind or air resistance could require refinement of the flight model; a different length–width relation changes the rectangle model.
- The acceleration 10 here is a stated simplification, not a replacement for a paper’s instructed value of g.
Find the rectangle area when its width t is 5 metres.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
A=t(t+2)=5×7=35 m².
Match each modelling requirement to its statement.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Coefficients carry the units of the stated parameter; a formula extrapolated past its interval is not evidence.
In your course
- AQA A-level Mathematics (7357), specification section C.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Parametric model: A model expressing two quantities using a common variable and stated assumptions.
- Name the parameter, its units and allowed interval.
- Watch for: the flight formula giving negative height after t=4 does not describe continued free flight below the ground.